Consider an open, linear system involving only circulating capital and producing \(n\) commodities by \(n\) industries of pure joint production. Furthermore, assume that (1) the input–output coefficients are fixed; (2) there are no non-competitive imports; (3) the net product is distributed to profits and wages that are paid at the end of the common production period; (4) the price of a commodity obtained as an output at the end of the production period is the same as the price of that commodity used as an input at the beginning of that period (‘stationary prices’); and (5) each process uses only one type of labor.

On the basis of these assumptions, the price side of the system is described by^{Footnote 5}

$${\mathbf{p}}^{\text{T}} {\mathbf{B}} = {\mathbf{w}}^{\text{T}} {\hat{\mathbf{l}}} + {\mathbf{p}}^{\text{T}} {\mathbf{A}}[{\mathbf{I}} + {\hat{\mathbf{r}}}]$$

(1)

where \({\mathbf{B}}\,( \ge {\mathbf{0}})\) denotes the *n *× *n* output coefficients matrix, \({\mathbf{A}}\,( \ge {\mathbf{0}})\) the *n *× *n* input coefficients matrix, **I** the *n *× *n* identity matrix, \({\hat{\mathbf{l}}}\,(l_{j} > 0)\) the *n *× *n* matrix of direct labor coefficients, \({\mathbf{p}}^{\text{T}}\) (\(> {\mathbf{0}}^{\text{T}}\)) the 1 × *n* vector of commodity prices, \({\mathbf{w}}^{\text{T}} \,(w_{j} > 0)\) the 1 × *n* vector of money wage rates, and \({\hat{\mathbf{r}}}\) (\(r_{j} \ge - 1\) and \({\hat{\mathbf{r}}} \ne {\mathbf{0}}\)) the *n *× *n* matrix of the exogenously given and constant sectoral profit rates.

Provided that \([{\mathbf{B}} - {\mathbf{A}}]\) is non-singular, Eq. (1) can be rewritten as

$${\mathbf{p}}^{\text{T}} = {\mathbf{w}}^{\text{T}} {\varvec{\Lambda}} + {\mathbf{p}}^{\text{T}} {\mathbf{H}}$$

(2)

where \({\mathbf{H}} \equiv {\mathbf{A}}{\hat{{\mathbf{r}}}}[{\mathbf{B}} - {\mathbf{A}}]^{ - 1}\) may be considered as the ‘\({\hat{\mathbf{r}}}\,-\) vertically integrated technical coefficients matrix’, and \({\varvec{\Lambda}} \equiv {\hat{\mathbf{l}}}[{\mathbf{B}} - {\mathbf{A}}]^{ - 1}\) denotes the matrix of direct and indirect labor requirements per unit of net output for each commodity.^{Footnote 6}

The quantity side of the system is described by

$${\mathbf{Bx}} = {\mathbf{Ax}} + {\mathbf{y}}$$

or

$${\mathbf{x}} = [{\mathbf{B}} - {\mathbf{A}}]^{ - 1} {\mathbf{y}}$$

(3)

and

$${\mathbf{y}} = {\mathbf{c}}_{w}^{{}} + {\mathbf{c}}_{p}^{{}} - {\mathbf{Im}} + {\mathbf{d}}$$

or, setting \({\mathbf{Im}} = {\hat{\mathbf{m}}\mathbf{Bx}}\),

$${\mathbf{y}} = {\mathbf{c}}_{w}^{{}} + {\mathbf{c}}_{p}^{{}} - {\hat{\mathbf{m}}\mathbf{Bx}} + {\mathbf{d}}$$

(4)

where \({\mathbf{x}}\) denotes the *n *× 1 activity level vector, \({\mathbf{y}}\) the vector of effective final demand, \({\mathbf{c}}_{w}^{{}}\) the vector of consumption demand out of wages, \({\mathbf{c}}_{p}^{{}}\) the vector of consumption demand out of profits, \({\mathbf{Im}}\) the import demand vector, \({\mathbf{d}}\) (\(\ge {\mathbf{0}}\)) the autonomous demand vector (government expenditures, investments and exports), and \({\hat{\mathbf{m}}}\) the matrix of imports per unit of gross output of each commodity.

If \({\mathbf{f}}\) (\(\ge {\mathbf{0}}\)) denotes the exogenously given, uniform and constant consumption pattern (associated with the two types of income), and \(s_{w}\, (s_{p} )\) denotes the savings ratio out of wages (out of profits), where \(0 \le s_{w} < s_{p} \le 1\), then Eqs. (2) and (3) imply that the consumption demands amount to

$${\mathbf{c}}_{w} = [(1 - s_{w} )({\mathbf{w}}^{\text{T}} {{\varvec{\Lambda} \mathbf{y}}})({\mathbf{p}}^{\text{T}} {\mathbf{f}})^{ - 1} ]{\mathbf{f}}$$

(5)

$${\mathbf{c}}_{p} = [(1 - s_{p} )({\mathbf{p}}^{\text{T}} {\mathbf{Hy}})({\mathbf{p}}^{\text{T}} {\mathbf{f}})^{ - 1} ]{\mathbf{f}}$$

(6)

where the terms in brackets represent the levels of consumption demands out of wages and profits, respectively.

Substituting Eqs. (5) and (6) into Eq. (4) finally yields

$${\mathbf{y}} = [{\mathbf{C}} - {\mathbf{M}}]{\mathbf{y}} + {\mathbf{d}}$$

(7)

where

$${\mathbf{C}} \equiv ({\mathbf{p}}^{\text{T}} {\mathbf{f}})^{ - 1} {\mathbf{f}}[(1 - s_{w} ){\mathbf{w}}^{\text{T}} {\varvec{\Lambda}} + (1 - s_{p} ){\mathbf{p}}^{\text{T}} {\mathbf{H}}]$$

is the matrix of total consumption demand, and

$${\mathbf{M}} \equiv {\hat{\mathbf{m}}\mathbf{B}}[{\mathbf{B}} - {\mathbf{A}}]^{ - 1}$$

is the matrix of total import demand.

Provided that \([{\mathbf{I}} - {\mathbf{C}} + {\mathbf{M}}]\) is non-singular (consider Mariolis 2008, pp. 660–661 and 663), Eq. (7) can be uniquely solved for \({\mathbf{y}}\):

$${\mathbf{y}} = {\mathbf{{\varvec{\Pi}} d}}$$

(8)

where \({\varvec{\Pi}} \equiv [{\mathbf{I}} - {\mathbf{C}} + {\mathbf{M}}]^{ - 1}\) is the static multiplier linking autonomous demand to net output, i.e., a matrix multiplier in a Sraffian joint production and open economy framework. It is a multiplier of commodities (instead of industries) and the multiplier effects depend, in a rather complicated way, on the: (1) technical conditions of production; (2) imports per unit of gross output; (3) distributive variables (\(w_{j}^{ - 1} {\mathbf{w}}\) and \({\hat{\mathbf{r}}}\)); (4) savings ratios out of wages and profits; (5) consumption pattern; and (6) physical composition of autonomous demand.^{Footnote 7} It goes without saying that, in general, any change in relative commodity prices, induced, directly or indirectly, by changes in income distribution, alters the elements of this matrix multiplier and, therefore, the total multiplier effects become ambiguous. This ambiguity is a distinctive feature of the multiplier process in Sraffian frameworks (Metcalfe and Steedman 1981; Mariolis 2008).

Finally, Eqs. (3) and (8) imply that the volumes of employment, \({\mathbf{L}} \equiv {\hat{\mathbf{l}}\mathbf{x}}\), associated with \({\mathbf{d}}\) are given by

$${\mathbf{L}} = {\varvec \Lambda}{\varvec \Pi} {\mathbf d}$$

(9)

Thus, the employment effects of \({\mathbf{d}}\) can be decomposed (Kahn 1931) into ‘primary employment’ effects, i.e.,

$${\mathbf{L}}_{\rm I} \equiv{\mathbf{{\varvec{\Lambda}} d}}$$

(9a)

and ‘secondary employment’ effects, i.e.,

$${\mathbf{L}}_{{{\rm I}{\rm I}}} \equiv {\mathbf{L}} - {\mathbf{L}}_{\rm I} = {\varvec{\Lambda}}[{\varvec{\Pi}} - {\mathbf{I}}]{\mathbf{d}}$$

(9b)

From Eqs. (8) and (9), it then follows that the changes on (1) the money value of net output, \(\Delta_{y}^{i}\) (output multiplier); (2) the money value of imports, \(\Delta_{{Im}}^{i}\) (import multiplier); and (3) total employment, \(\Delta_{L}^{i}\) (total employment multiplier), induced by the increase of 1 unit of the autonomous demand for commodity *i*, are given by

$$\Delta_{y}^{i} \equiv {\mathbf{p}}^{\rm T} {\mathbf{\varvec{\Pi} e}}_{i}$$

(10)

$$\Delta_{{Im}}^{i} \equiv {\mathbf{p}}^{\text{T}} {{\mathbf M}{\varvec \Pi} {\mathbf e}}_{i}$$

(11)

and

$$\Delta_{L}^{i} \equiv {\mathbf{e}}^{\text{T}} {{\varvec\Lambda} {\varvec \Pi} {\mathbf e}}_{i}$$

(12)

respectively.